How should one read the s*(t) function in Mendelson's Introduction to Mathematical Logic?

Question

I'm self-teaching logic and doing it by means of following Elliot Mendelson's Introduction to Mathematical Logic (6th edition). In p.56 he defines a a function s* which, in his words, 'assigns to each term t of L an element s*(t) in the domain D'.

For variables and individual constants, I understand the definition. But things get a bit sketchy when it comes to the function letters.

He says: If $f_k^n$ is a function letter, $(f_k^n)^M$ is the corresponding operation in D, and $t_1$, ... , $t_n$ are terms, then

s* $f^n_k$($t_1$, ..., $t_n$)) = ($f^n_k)$$^M$ (s*(t$_1$),...,s*($t_n$))

This function is used to define the notion of satisfiability (which, in turn, is used to define truth in Tarskian terms) and I'm having some problem understanding it. If anyone who's familiar with the text could help me I'd be very thankful.

*n is the number of arguments taken by the function letter, and k is simply an index number

Answer 1

We have an interpretation with domain $D$ that gives to the "basic" syntactical objects of the language $\mathcal L$ a "meaning" :

• to every constant symbol $a_i$ a denotation $(a_i)^M$, that is an objcet of $D$

• to every predicate letter $A_j^n$ an $n$-place relation in $D$

• to every function letter $f_j^n$ an $n$-place operation in $D$.

A variable assignment function $s$ assign to every variabke $x_i$ an object $s(x_i) \in D$, i.e. a sort of "temporary" denotation.

With $s$ we can extend this "temporary" denotation to all terms with variables inside : this is the "gist" of $s^*$.

Consider e.g. the first-order language of arithmetic with two constants $0$ and $1$ and two opeartion symbols : $+$ and $\times$ and the usual interpretation with domain $\mathbb N$.

The denotation of the two constants are : the numbers zero and one, respectively, while to the two function symbols are assigned the operations of sum and product between natural numbers, respectively.

The expression :

$x_1+0$

is a term with a variable inside and thus, by himself, it has no denotation.

Consider now the variable assignment function $s$ such that :

$s(x_1)=0, s(x_2)=1, \ldots$.

What is :

$s^*(x_1+0)$ ?

It is the object of $\mathbb N$ (i.e. a natural number) "calculated" applying the operation of sum to the number $0$ (i.e. $s(x_1)$) with itself, i.e.

$0+0=0$.

If instead we consider the new term $x_2+0$, the same $s$ "induces" a denotation $s^*(x_2+0)=1$, because $s(x_2)=1$.